You might think most people who pass math classes learned what the class is about, but that's not correct. People pass by learning slightly less than the bare minimum and cheating.

I don't think the methodology is the issue with this one. 500 people can absolutely be a legitimate sample size. Under basic assumptions about the sample being representative and the effect size being sufficiently large you do not need more than a couple hundred participants to make statistically significant observations. 54% being close to 50% doesn't mean the result is inconclusive. With an ideal sample it means people couldn't reliably differentiate the human from the bot, which is presumably what the researchers believed is of interest.

In a hypothetical and highly unlikely world where everyone had to pay Oracle to use Java, everyone would switch to something else. It would be guaranteed suicide. Anyway, in that world, they would need to both make this ridiculous decision and win an unwinnable legal battle afterwards. It's not a realistic concern.

One definition of the complex numbers is the set of tuples (x, y) in R^(2) with the operations of addition: (a,b) + (c,d) = (a+c, b+d) and multiplication: (a,b) * (c,d) = (ac - bd, ad + bc). Then defining i := (0,1) and identifying (x, 0) with the real number x, we can write (a,b) = a + bi.

"Measure" is meant in the specific sense of measure theory. The prototypical example is the Lebesgue measure, which generalizes the intuitive definition of length, area, volume, etc. to N-dimensional space.

As a pseudo definition, we may assume:

1. The measure of a rectangle is its length times its width.

2. The measure of the disjoint union of two sets is the sum of their measures.

In 2), we can relax the assumption that the two sets are disjoint slightly, as long as the overlap is small (e.g. two rectangles overlapping on an edge). This suggests a definition for the measure of any set: cover it with rectangles and sum their areas. For most sets, the cover will not be exact, i.e. some rectangles will lie partially outside the set, but these inexact covers can always be refined by subdividing the overhanging rectangles. The (Lebesgue) measure of a set is then defined as the greatest lower bound of all possible such approximations by rectangles.

There are 2 edge cases that quickly arise with this definition. One is the case of zero measure: naturally, a finite set of points has measure zero, since you can cover each point with a rectangle of arbitrarily small area, hence the greatest lower bound is 0. One can cover any countably infinite set with rectangles of area epsilon/n^(2) so that the sum can be made arbitrarily small, too. Even less intuitively, an uncountably infinite and topologically dense set of points can have measure 0 too, e.g. the Cantor set.

The other edge case is the unmeasurable set. Above, I mentioned a subdivision process and defined the measure as the limit of that process. I took for granted that the limit exists. Indeed, it is hard to imagine otherwise, and that is precisely because under reasonably intuitive axioms (ZF + dependent choice) it is consistent to assume the limit always exists. If you take the full axiom of choice, you may "construct" a counterexample, e.g. the Vitali set. The necessity of the axiom of choice in defining this set ensures that it is difficult to gain any geometric intuition about it. Suffice it to say that the set is both too "substantial" to have measure 0, yet too "fragmented" to have any positive measure, and is therefore not well behaved enough to have a measure at all.

Sometimes I update and can no longer boot so I go outside, does that counf

ASD coded, not literally ASD. The thematic struggle with social convention counts IMO. But Data has no self loathing or whatever. He just wants to understand. We stan Data.

No such thing has been "mathematically proven." The emergent behavior of ML models is their notable characteristic. The whole point is that their ability to do anything is emergent behavior.

What is this article supposed to show?

Don't use boiling water. 195-205F for black tea. Brew time typically varies by preference from 3-4 minutes, but 5 isn't terrible.

Mathematical constructivists hate this meme

I wonder if this actually fixes the ancient dwm bug that causes simultaneous motion on multiple monitors with different refresh rates to make the whole window manager choppy. That bug has existed since at least Vista, and it sucks. Nothing like buying a 240Hz monitor and not being able to watch videos on my secondary one without bringing them both down to what looks like 60.